# Perform simple regression analysis and calculate Durbin-Watson statistic

A freshly brewed shot of espresso has three distinct components: the heart, body, and crema. The separation of these three components typically lasts only 10 to 20 seconds. To use the espresso shot in making a latte, a cappuccino, or another drink, the shot must be poured into the beverage during the separation of the heart, body, and crema. If the shot is used after the separation occurs, the drink becomes excessively bitter and acidic, running the final drink. Thus, a longer separation time allows the drink-maker more time to pour the shot and ensure that the beverage will meet expectations. An employee at a coffee shop hypothesized that the harder the espresso grounds were tamped down into the portafilter before brewing, the longer the separation time would be. An experiment using 24 observations was conducted to test this relationship. The independent variable Tamp measures the distance, in inches, between the espresso grounds and the top of the portafilter (i.e., the harder the tamp, the greater the distance). The dependent variable Time is the number of seconds the heart, body, and crema are separated (i.e., the amount of time after the shot is poured before it must be used for the customer's beverage). The data are stored in Espresso.

a. Use the least-squares method to develop a simple regression equation with Time as the dependent variable and Tamp as the independent variable.

b. Predict the separation time for a tamp distance of 0.50 inch.

c. Plot the residuals versus the time order of experimentation. Are there any noticeable patterns?

d. Compute the Durbin-Watson statistic. At the 0.05 level of significance, is there evidence of positive autocorrelation among the residuals?

e. Based on the results of ( c ) and ( d ), is there reason to question the validity of the model?

https://brainmass.com/statistics/regression-model-validation/perform-simple-regression-analysis-and-calculate-durbin-watson-statistic-590266

#### Solution Preview

Use the least-squares method to develop a simple regression equation with Time

as the dependent variable and Tamp as the independent variable.

Answer: we run the regression analysis in excel. I include the excel file. The regression equation is Time=17.08-5(Tamp)

b. Predict the separation time for a tamp distance of 0.50 inch.

Answer: when tamp=0.50, plug into regression equation to get ...

#### Solution Summary

The solution gives detailed steps on performing simple regression analysis, calculating Durbin-Watson statistic and checking validity of model using Excel.

The solution gives detailed discussion on multiple regression after obtaining the software output.

Alice wanted to examine the impact of family ownership on the valuation and/or performance of newspaper publishing companies in Hong Kong for her research project. She decided to download data from a publicly available database on the Internet and base her research entirely on secondary data.

When browsing the Internet, Alice found that information about family ownership of newspaper publishing companies in Hong Kong was available for 2003, while the relevant financial statements could be obtained for every year from 1990 to 2004. Her total sample comprised 10 firms. Following an initial analysis, Alice decided to use return on assets as a measure of each newspaper publishing company's performance. This was calculated as the ratio of earnings before interest and taxes and average assets using the formula:

return on assets = ( Earnings before interest and taxes) / (average assets)

Her independent variable was the level of family ownership, defined as the percentage of the newspaper publishing company owned by the family. Alice presented her research ideas as a model using the regression equation:

Equation 1: ROA t = α + β Ownership t

Where:

ROA t = return on assets

Ownership t = percentage of newspaper publishing company owned by family

α and β = intercept and slope coefficients

After entering the data she had obtained for the 10 companies for 2003 in her statistical analysis software, Alice performed a regression analysis. This analysis calculated values for the intercept and slope coefficients for her regression equation as well as providing the following statistics:

Equation 2: ROA t = 0.05 + (0.002)(Ownership t)

Probability (p) for α: 0.01

Probability (p) for β: 0.15

R2: 0.02

Durbin-Watson statistic: 1.1

F-Statistic: 0.1

Based on these statistics, Alice felt that the value of the estimate of the beta coefficient was both economically and statistically significant. Alice's good friend, Betty, agreed to take a look at the model and make suggestions regarding the inclusion of independent variables. After looking at the model Betty asked, "Why don't you examine this relationship over a longer period of time, rather than just for the one year. In addition, you don't seem to have included any control variables in your regression model. I think it is generally advisable to control for firm size, industry affiliation, and other factors that.

reflect a change in capital structure and future growth potential in order to single out the impact of ownership."

Alice agreed to follow the advice of Betty and re-examine the model. She discovered that she could obtain ownership data for 2002 if she contacted the Hong Kong Stock Exchange. However, she felt this would not provide sufficient data for longitudinal analysis. At the same time she found data on four more publishing companies, bringing the total sample to 14.

She decided to measure firm size using data on earnings before interest and taxes that she had already collected and used to calculate return on assets. She justified this to Betty, saying, "Earnings before interest and taxes seems to be a great proxy for firm size; big firms will have big earnings." She also decided to include data on marketable securities, commenting, "These indicate a company with high liquidity and the fact that it may not be in need to further borrow funds.

Their leverage will decline as the amount of marketable security increases." Based upon her discussion with Betty, Alice presented her research idea to her professor, presenting Equations 1 and 2 and the following model:

Equation 3: ROA t = α + β 1 Ownership t + β 2 EBIT t + β 3 Market + β 4 Dummy

Where:

ROA t = return on assets, calculated by dividing earnings before interest and taxes by average assets

Ownership t = percentage on newspaper publishing company owned by family

EBIT t = earnings before interest and taxes

Market = amount of marketable securities in millions of Hong Kong dollars

Dummy = dummy variable for industry. It equals one if a company is part of newspaper publishing sector, otherwise zero

α and β = intercept and slope coefficients

Questions

1. After Alice had explained her model to her professor, she was asked, "What problems can you identify in testing the initial model in Equation 1 with your data?" Provide a response for Alice.

2. Alice's next question focused on the analysis associated with Equation 2. "Why do you believe that the 0.02 coefficient estimate is both economically and statistically relevant?" Provide a response for Alice.

3. In the meeting Alice had with her academic advisor she was told the inclusion of the dummy and control variables in Equation 3 adds nothing to the research. Why do you think her professor made this statement?